golden ratio on the 200th term?

Question

what problems arise when we try to find the 200th term of the sequence

An+1=1+1/An

when A1=1+1
A2=1+ 1/1+1 etc.....

i can't find the problem that's supposedly there.... any help please?

Answer 1

Your notation is a little confusing. Is this your sequence:
a_(n+1) = 1/a_n, (where _ indicates subscripts) and a_1 = 2?? If that is your problem, then a_n = (f_(n+2))/(f_(n+1)), where f_n is the n-th Fibonacci number. We know that f_n = (u^n - v^n)/sqrt(5), where u = (1 + sqrt(5))/2 and v = (1 - sqrt(5))/2.

You can easily prove by induction that a_n = (f_(n+2))/(f_(n+1)), because
f_(n+2) = f_(n+1) + f_n.